What is the value of P if `log_(e)2.log_(p)625=log_(10)16.log_(e)10` ?

Options

A. 2
B. 4
C. 5
D. 7

Answer

To solve the equation \( \log_{e}2 \cdot \log_{p}625 = \log_{10}16 \cdot \log_{e}10 \), we will follow these steps:

### Step 1: Rewrite the logarithms
First, we can express \( \log_{p}625 \) and \( \log_{10}16 \) using the change of base formula:
\[ \log_{p}625 = \frac{\log_{e}625}{\log_{e}p} \]
\[ \log_{10}16 = \frac{\log_{e}16}{\log_{e}10} \]

### Step 2: Substitute into the equation
Substituting these into the original equation gives:
\[ \log_{e}2 \cdot \frac{\log_{e}625}{\log_{e}p} = \frac{\log_{e}16}{\log_{e}10} \cdot \log_{e}10 \]
This simplifies to:
\[ \log_{e}2 \cdot \frac{\log_{e}625}{\log_{e}p} = \log_{e}16 \]

### Step 3: Isolate \( \log_{e}p \)
Rearranging the equation to isolate \( \log_{e}p \):
\[ \log_{e}2 \cdot \log_{e}625 = \log_{e}16 \cdot \log_{e}p \]
\[ \log_{e}p = \frac{\log_{e}2 \cdot \log_{e}625}{\log_{e}16} \]

### Step 4: Calculate \( \log_{e}625 \) and \( \log_{e}16 \)
We know that:
\[ 625 = 5^4 \implies \log_{e}625 = 4\log_{e}5 \]
\[ 16 = 2^4 \implies \log_{e}16 = 4\log_{e}2 \]

### Step 5: Substitute back into the equation
Substituting these values back into the equation gives:
\[ \log_{e}p = \frac{\log_{e}2 \cdot 4\log_{e}5}{4\log_{e}2} \]
The \( 4\log_{e}2 \) cancels out:
\[ \log_{e}p = \log_{e}5 \]

### Step 6: Solve for \( p \)
Since \( \log_{e}p = \log_{e}5 \), we can conclude that:
\[ p = 5 \]

Thus, the value of \( P \) is \( 5 \).

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